Reality and Virtual Reality in Mathematics

In this talk I introduce three of the twentieth century’s main philosophies of mathematics and argue that of those three, one describes mathematical reality, the “reality”of the other two being merely virtual. What are mathematical objects, really? What, for example, is that thing that we call “the number one”, or “the set of all positive whole numbers”, or “the shortest path between two points on the surface of a sphere”? Most mathematicians (let alone most people) would …nd little interest in such questions, since they are totally preoccupied with the practice of their discipline rather than with questions about its meaning. In this talk I shall outline three of the standard philosophical approaches to the meaning of mathematics and present a case that one of those three represents the reality of mathematics, each of the other two amounting to virtual reality. (I should add that there is a fourth standard approach, known as logicism, in which mathematics is regarded as, or reduced to, the formal, axiomatic theory of logical propositions. This philosophy, advocated especially by Gottlob Frege, Bertrand Russell and A.N. Whitehead [12], is, from the viewpoint of meaning at least, similar to so-called formalism that will be discussed later, in that mathematics— regarded as an extension of logic— has form without content.) The …rst approach that I want to discuss is known as platonism. The platonist mathematician believes that mathematical objects do exist, in perfect “forms”, and that what mathematicians actually work with are, in Plato’s vivid metaphor, mere shadows cast by those perfect forms on the wall of the mathematical cave in which our intellects are con…ned. For the platonist, the number we call “one” is a real object, of which we work only with imperfect representatives (on paper or chalkboard or in the mind’s eye). Likewise, there is a perfect form of the sphere,